Potential diffusion processes of real-world systems are relevant to the underlying network structure and dynamical mechanisms. The vast majority of the existing work on spreading dynamics, in response to a large-scale network, is built on the condition of the infinite initial state, i.e., the extremely small seed size. The impact of an increasing seed size on the persistent diffusion has been less investigated. Based on classical epidemic models, this paper offers a framework for studying such impact through observing a crossover phenomenon in a two-diffusion-process dynamical system. The two diffusion processes are triggered by nodes with a high and low centrality, respectively. Specifically, given a finite initial state in networks with scale-free degree distributions, we demonstrate analytically and numerically that the diffusion process triggered by low centrality nodes pervades faster than that triggered by high centrality nodes from a certain point. The presence of the crossover phenomenon reveals that the dynamical process under the finite initial state is far more than the vertical scaling of the spreading curve under an infinite initial state. Further discussion emphasizes the persistent infection of individuals in epidemic dynamics as the essential reason rooted in the crossover, while the finite initial state is the catalyst directly leading to the emergence of this phenomenon. Our results provide valuable implications for studying the diversity of hidden dynamics on heterogeneous networks.